Distance formula: Between (1, 4) and (5, 10) Between (3, 8) and 3, 11) Sketch and write the equation of the following:

Transcription

1 Distance formula: Between (1, 4) and (5, 10) Between (3, 8) and 3, 11) Sketch and write the equation of the following: locus of all points (x,y) that are 5 units from (2,3) locus of all points 3 units from y = 2 locus of all points equidistant from (4,6) and (8, 12) locus of all points twice as far from (0, 0) as from (6,0) locus of all points as far from (0, 4) as from y = -2

2 locus of points 2 units from x = 3 and 4 units from (3, 5) locus of all points the sum of whose distance from (-4, 0) and (4, 0) is 10 locus of all points 2 units closer to (0, 3) than to (0, -3) locus of all points 4 units from (0, 0) and equidistant from (0,0) and (0,6)

3 Write in standard form (center-radius): 2x 2 + 2y y = 1 (general form/whoopie form)

4 Write the equation in standard form: Center is ( 2,3), circle passes through (5,6) Diameter endpoints are ( 8,0) and ( 0,6) The center is ( 3,1) and tangent to x=4 Contains ( 2,16) and has x-intercepts 2 and 32 Graph the semi circles: a. y = 9 x 2 b. y = 9 x 2 c. x = 5 16 y 2 d. y = 16 ( x 5) 2

5 Solve the system: x y = 3, x 2 + y 2 10x + 4y = 13 Find the equation of the circle through L(8,2), M(1,9), and N(1,1). a. With perp bis. b. Solving a system How about through (7,5), (1, 7), and (9, 1) Describe the set of points: x 2 + y 2 + 2x + 2y + 2 = 0 x 2 + y 2 6x + 8y + 26 = 0

6 ( x 2 + y 2 1) ( x 2 + y 2 4) = 0 Find the equation of the line passing through the points (6,3) and (1,0) and write in a. point-slope form b. slope-intercept form c. standard form Find the area of the circle: ( x 12) 2 + y + 5 ( ) 2 = 49 Find the equation of the line that has a positive slope and is tangent to the circle ( x 1) 2 + ( y 1) 2 = 4 at one of its y-intercepts. Write in slope-intercept form. Determine the angle of inclination of:

7 y = 3x + 4 Determine the angle of inclination of: y = 5x +1 A line has an x-intercept of 5 and is perpendicular to a line who s inclination is 130º. Find the equation of the line. Find the dinstance from ( 2, 3) to y = 4x +1 From the point (7. 1), tangent lines are drawn to the circle ( x 4) 2 + ( y 3) 2 = 4. Find the slopes of these lines. Find the distance between 3x + 4y = 12 and 3x + 4y = 24.

8 ELLIPSES Locus definition: the locus of points such that the sum of the distances to two fixed points (foci) is constant (length of the major axis). Key terms: center, vertices (refer to the major axis vertices), major axis, minor axis, foci Expanded form: Standard form: ( x h) 2 + y k a 2 ( )2 b 2 = 1 horizontal major axis or ( x h) 2 b 2 + ( y k)2 a 2 = 1 vertical major axis

9 where a=distance from center to major vertices b=distance from center to minor vertices c=distance from center to foci Relationship between a, b, and c: a2 b 2 = c 2 a 2 = b 2 + c 2 Why? Sketch ( x 3) ( y 6)2 25 = 1 Find the location of the foci of:

10 Find the equation of the ellipse: Vertices are (5,9) and (5,1) and one focus is (5,7). 1) Equation: x y2 9 = 1 a= b= c= Center: Endpts of Major axis: and Endpts of Minor axis: and Foci: and 2) Equation: (x 2)2 4 + (y + 4)2 16 = 1 a= b= c= Center: Endpts of Major axis: and Endpts of Minor axis: and Foci: and

11 3) Equation: a= b= 3 c= Center: (0,0) Endpts of Major axis: (0, 9) and Endpts of Minor axis: and Foci: and 4) Equation: a= b= c= Center: (4, 2) Endpts of Major axis: and Endpts of Minor axis: (8, 2) and Foci: (4,1) and 5) Equation: a= b= c= Center: Endpts of Major axis: and Endpts of Minor axis: and Foci: and

12 6) Equation: (x 3) (y + 1) 2 = 1 a= b= c= Center: Endpts of Major axis: and Endpts of Minor axis: and Foci: and 7) Equation: 4(x 1) y 2 = 144 a= b= c= Center: Endpts of Major axis: and Endpts of Minor axis: and Foci: and 8) Equation: a= b= c= Center: (3,7) Endpts of Major axis: (8,7) and Endpts of Minor axis: and Foci: (6,7) and

13 9) Equation: a= b= c= Center: Endpts of Major axis: and Endpts of Minor axis: (1,4) and (7,4) Foci: (4,8) and 10) Equation: 9x 2 + y x 6y + 9 = 0 a= b= c= Center: Endpts of Major axis: and Endpts of Minor axis: and Foci: and 11) Equation: 4x 2 + 3y x 18y + 31 = 0 a= b= c= Center: Endpts of Major axis: and Endpts of Minor axis: and Foci: and

14 Thursday: 1. Bonus quiz on hyperbolas 2. Hyperbolas notes 3. HW due Wednesday pg 235, 237 #1-21, ODD hyperbola packet Bonus quiz: Find the location of the foci of ( x 1) 2 4 ( y + 3)2 25 = 1 Locus definition of a hyperbola: The locus of points such that the difference of the distances to two fixed points (foci) is constant (2a)

15 Key terms: center, vertices, asymptotes, foci, transverse axis (major), conjugate axis (minor), fundamental rectangle Expanded form: Standard form: ( x h) 2 y k a 2 ( )2 b 2 = 1 horizontal transverse (major) axis or ( y k) 2 a 2 ( x h)2 b 2 = 1 vertical transverse (major) axis where a=distance from center to vertices b= half the conjugate axis (used to create the fund. rectangle) c=distance from center to foci

16 Relationship between a, b, and c: a 2 + b 2 = c 2 (c is biggest) Find coordinates of the foci, vertices, and equation of the asymptotes: ( y + 5) 2 x = 1 Find coordinates of the foci, vertices, and equation of the asymptotes: x 2 4y 2 2x +16y 19 = 0 Find the equation of the hyperbola: Vertex at (0, 12) and a focus at (0, 13). A vertex at (8,0) and an asymptote with equation y = 1 2 x Vertices are (4,0) and (4,8) and one asymptote has a slope of 1. Sketch the hyperbola and asymptotes. Give equations of the asymptotes.

17 xy = 8 xy = 8 ( x 2) ( y + 4) = 8 Locus proof: Find the equation of the locus of points such that the difference of the distances to ( 5,0) and (5,0) is always 6. Tuesday 1. More hyperbola examples 2. HW due Wednesday: pg 235, 237 #1-21, ODD hyperbola packet 1) Equation: x2 9 y2 16 = 1 a= b= c= Center: Vertices: and Equation of asymptotes: Foci: and

18 2) Equation: (y 2)2 4 x2 9 = 1 a= b= c= Center: Vertices: and Equation of asymptotes: Foci: and 3) Equation: a= b= c= Center: Vertices: (1,4) and (1,-2) Equation of asymptotes: Foci: (1,6) and

19 4) Equation: a= b=5 c= Center: ( 2, 3) Vertices: ( 4, 3) and Equation of asymptotes: Foci: and 5) Equation: 16x 2 4y 2 = 64 a= b= c= Center: Vertices: and Equation of asymptotes: Foci: and 6) Equation: 9x x 16y 2 32y 79 = 0 a= b= c= Center: Vertices: and Equation of asymptotes: Foci: and

20 7) Equation: 7x 2 6y 2 = 54 a= b= c= Center: Vertices: and Equation of asymptotes: Foci: and 8) Equation: a= b= c= Center: Vertices: (1,4) and (1,0) Equation of asymptotes: y 2 = ± 1 (x 1) 3 Foci: and 9) Equation: a= b=1 c= Center: Vertices: and Equation of asymptotes: Foci: and *Draw the asymptotes the best you can

21 10) Equation: a= b= c= Center: Vertices: and Equation of asymptotes: Foci: and *Asymptotes are drawn in Find the eccentricity of each:

22 Wednesday: 1. Hyperbola-locus proof 2. Parabola-proof of a = 1 4 p 3. Check/review hyperbola hw 4. Computers 5. Bonus-respond to - how s it going? 6. HW due Thursday -eccentricity of conic sections worksheet ODDS

23 Hyperbola Locus proof: Find the equation of the locus of points such that the difference of the distances to ( 5,0) and (5,0) is always 6. Parabola locus proof: Find the locus of points equidistant from y = p and ( 0, p). General form of a conic: Assumes not degenerate Some grapher examples:

24 Example: Example: 5. HW due Monday: pg. 250 #1-13 ODD Sketch. Most are degenerate conics. x 2 + y 2 = 0 this is a degenerate... x 2 + y 2 = 1 this is a degenerate... x 2 y 2 = 0 this is a degenerate... x 2 y 2 = 1 x 2 y 2 2x 4y 3 = 0 this is a degenerate... (x 1) 2 + ( y 3) 2 = 0 this is a degenerate... (x 1) 2 + ( y 3) 2 = 1 this is a degenerate... 5(x 1) 2 + 2( y 3) 2 = 0 this is a degenerate...

25 x 2 = 4 this is a degenerate... Parabola: two parallel lines, two coinciding lines, a single line

26 Generic focus/directrix definition for every conic: The locus of points such that the ratio of the distances to a fixed point and a line is always constant. We call that ratio the eccentricity ellipse: 0<e<1 parabola e=1 hyperbola e>1

27 Find the locus of points such that the ratio of the distances to (0,3) and y = 12 is 1:2.

28 Monday: 1. Systems of quadratic equations Future Polar form of conics rotated conics: Chapter%2010/10-5%20notes%20day2.pdf You ll need this: cot 2θ = 3 7, find cosθ and tanθ Why we can always rotate the axis between 0º and 90º?

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30 Example 1-just a point Angle of rotation from x-axis to x' -axis is 45º. If (2,0) is a point with respect to the x -y plane, what are the coordinates with respect to the x-y plane? This is strange... These let you go from rotated points to regular points. Yet, these let us go from a regular equation to a rotated equation. To go in the other direction: x' = x cosθ + ysinθ y' = xsinθ + ycosθ We just don t really need these equations to graph our conics.

31 Example 2-the most basic rotated conic Write xy = 1 in terms of the x -y coordinate system. Calculate the vertices, foci, and asymptotes. Example 3-well, at least the angle is nice... Graph 2x2 + 3xy + y 2 = 2 Example 4- it just got real x 2 + 4xy 2y 2 = 6 Example 5-16x 2 24xy + 9y x 20y +100 = 0 Example 6-you know how this will turn out, right? x 2 + 4xy + 4y 2 = 0 The proofs: 1. Relating P(x,y) to P(x y ) to get

32 x' = x cosθ + ysinθ y' = xsinθ + ycosθ 2. Solving the system in part 1 to get x = x'cosθ y'sinθ y = x'sinθ + y'cosθ 3. Taking a conic Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 and substituting x = x'cosθ y'sinθ to put in the form y = x'sinθ + y'cosθ A'x' 2 + B'x'y'+ C 'y' 2 + D'x'+ E 'y'+ F ' = 0 -you are solving for the new coefficients in terms of the original coefficients A( x'cosθ y'sinθ ) 2 + B( x'cosθ y'sinθ )( x'sinθ + y'cosθ ) + C( x'sinθ + y'cosθ ) 2 + D( x'cosθ y'sinθ ) + E( x'sinθ + y'cosθ ) + F = 0 Ax' 2 cos 2 θ 2Ax'y'cosθ sinθ + Ay' 2 sin 2 θ + Bx' 2 sinθ cosθ By' 2 sinθ cosθ By'x'sin 2 θ + By'x'cos 2 θ + Cx' 2 sin 2 θ + Cy' 2 cos 2 θ + 2Cx'y'cosθ sinθ + Dx'cosθ Dy'sinθ + Ex'sinθ + Ey'cosθ + F = 0 x' 2 Acos 2 θ + Bsinθ cosθ + C sin 2 θ ( ) + y' ( Dsin+ E cosθ ) + F = 0 ( ) + x'y' ( 2Acosθ sinθ Bsin 2 θ + Bcos 2 θ + 2C cosθ sinθ ) + y' 2 ( Asin 2 θ Bsinθ cosθ + C cos 2 θ ) + x' Dcosθ + E sinθ ( 2Acosθ sinθ Bsin 2 θ + Bcos 2 θ + 2C cosθ sinθ )

33 A' = Acos 2 θ + Bsinθ cosθ + C sin 2 θ B' = 2( C A)cosθ sinθ + B cos 2 θ sin 2 θ ( ) C ' = Asin 2 θ Bsinθ cosθ + C cos 2 θ D' = Dcosθ + E sinθ E ' = Dsin+ E cosθ F ' = F 4. Set the B = 0 so that we can put the x -y conic in STANDARD FORM -we end up getting cot 2θ = A C B ( ) = 0 2( C A)cosθ sinθ + B cos 2 θ sin 2 θ ( C A)sin2θ + Bcos2θ = 0 Bcos2θ = ( A C)sin2θ cot 2θ = A C B 5. Show that B 2 4AC = B' 2 4A'C ' a. Show that A + C = A'+ C ' b. Show that A' C ' = ( A C)cos2θ + Bsin2θ c. Show that B' = Bcos2θ ( A C)sin2θ d. square and add the equations in part b and c to get ( A' C ') 2 + B' 2 = ( A C) 2 + B 2 e. square the equation in part a f. do part d e to show that B' 2 4A'C ' = B 2 4AC g. what does it all mean!?

34 Formulas: To find angle of rotation that will eliminate the xy term: cot( 2θ ) = A C B Converting rotated coordinates to regular x-y coordinates x = x'cosθ y'sinθ y = x'sinθ + y'cosθ Steps: 1. Find the angle that will eliminate the x y term: cot( 2θ ) = A C B 2. Plug into x = x'cosθ y'sinθ y = x'sinθ + y'cosθ 3. Plug those x and y values into the original equation 4. Do some algebra -the x y eliminates -put into standard form 5. Find important locations in terms of x -y 6. convert those important values back to x-y

35 Monday: 1a. New topic-complex # s 1. Review test problems and a few others 2. Test scale 3. Factors that could make a conics test more difficult 4. Last grade of quarter-quiz -rotation directrix of ellipse and hyperbola -circle through 3 points -focal radii length -systems of quadratic equations (2 methods) 5. HW due Wednesday pg. 875 #1-67 odd Complex Numbers in Polar Form Define Complex Number -has a real and imaginary part a+bi Argand Diagram/Complex Number Plane

36 *Remember, a complex number is a single number. It s unique that we are graphing it in a two dimensional plane. We tend to use the variable z to represent a complex number. Polar representation of a complex number:

37 z = r( cosθ + isinθ ) or rcisθ *r is called the modulus and is ALWAYS positive (unlike with polar coordinates) *θ is called the argument and is the same in both functions z = r = a 2 + b 2 denoted absolute value of z θ is the polar angle of z Converting back and forth: a = r cosθ b = rsinθ a 2 + b 2 = c 2 tanθ = b a

38 Convert 2( cos50º +isin50º ) to rectangular. *two methods Convert 5cis270º to rectangular Convert 1 2i to polar. Convert 5 to polar form in radians Let s take a second and note the major difference between: -polar form of (x,y) coordinates (r,theta) -polar form of a complex number z = a + bi Put each into polar form and graph: z 1 = 3+ 3i z 2 = 1+ i 3 z 1 z 2 = Product rule for complex numbers:

39 ( r 1 cisθ 1 )( r 2 cisθ 2 ) = r 1 r 2 cis( θ 1 +θ 2 ) Multiply 2 cos10º +isin10º ( )i 5 cos100º +isin100º ( ) Power rule for complex numbers- De Moivre s Theorem: If z = rcisθ, then z n = ( rcisθ ) n = r n cis( nθ ) Let s race: ( 2 i 2 ) 5 *two methods Evaluate ( 1 3i) 6 using De Moivre s. Should your answer have decimals? A random reminder: Conjugate of a Complex Number The conjugate of a + bi is a bi

40 Some notation: If z = a + bi, Re(z)=a and Im(z)=b 3-2i Re(3-2i)=3 Bonus quiz: Find the cube roots of 27i (in rectangular form of course). 1. re iθ = rcisθ identity 2. Conics quiz preview 3. Finding roots of complex numbers 4. Look over conics test -scale -accuracy analysis 5. HW due Thursday -pg. 875 #1-67 odd Euler s Formula: Example: Convert 3 8i to re iθ form. 73e 4.35i Roots of Complex Numbers:

41 1. Graph and find the three cube roots of 8cis90º. Namely, solve z 3 = 8. How many degrees are the roots apart? What type of polygon do the roots form? 2. Bonus quiz: Find the cube roots of 27i (in rectangular form of course). How many degrees are the roots apart? What type of polygon do the roots form? 3. Find the four fourth roots of 16. How many degrees are the roots apart? What type of polygon do the roots form? 4. Solve z 4 = i How many degrees are the roots apart? What type of polygon do the roots form? 5.

42 Solve z 3 = 4 + 2i ( 3+ i) 7 How many degrees are the roots apart? What type of polygon do the roots form? 6. Roots of unity- Find the 5th roots of unity. Product rule proof: ( cosθ 1 + isinθ 1 )( cosθ 2 + isinθ 2 ) Review of the Conics Test and Problems for the next Quiz A couple more terms worth knowing: Focal chord Focal width

43 New for quiz! Directrix given standard form equation. Proof of directrix of ellipse or hyperbola centered at the origin being x = ± a e *this is for horizontal opening. Same logic applies for vertical. Find the directrices of an ellipse a=4, b=3, c = 7 x y2 9 = 1 x = ± Hyperbola: foci are ( ±3,0 ) and directrices are x = ±1. Equation? More practice in 11.6 New for quiz! Circle through 3 points. Find an equation of the circle passing through the three points: A(4, 1), B(-2, 3), C(-8, -3). New for quiz! Tangent lines. Find the equation of the line tangent to the parabola: y 2 8x = 0 at the point (6,-3). read section 11.3 for more help

44 y 2 = 8x, ( 8, 8) The ellipse x 2 + 3y 2 = 57 at point ( 3,4 ) can be done the same way, but is messy. There is a formula : * x 1 x a 2 ( x 1, y 1 ). + y 1 y b 2 = 1 is the line tangent to the ellipse x 2 a 2 + y2 b 2 = 1 at New for quiz! Focal radii. Give the lengths of the focal radii F1P and F2P for the ellipse x 2 + 3y 2 = 28 and the point P(1, -3). Definition of a Log

45 Common bases: log x = log 10 x ln x = log e x Properties of Logs *#3 should be obvious from #1 *careful log( 2x 3 ) 3log( 2x) Change of Base: *quick proof

46 log a b = log c b log c a Other: log b a = log b c ( a c ) b log b x = *two reasons 1 log a b = log b a *from change of base Sketch of y = 2 x. Domain? Range? Sketch of y = log 2 x Domain? Range?

47 4. a) Find the equation of the exponential function f(x) if f(2) = 18, f(5) = 486 b) What is the value of x for which: f(x) = 54 f(x) = 2 9

48 1. f(x) = x 1 x f(x) = x x f(x) = x 4 (x 1)(x + 2) 4. f(x) = 3x2 x 2 3x 5. f(x) = x 2 16 x 2 6x f(x) = x2 x 2 x 2 2x + 1

49 7. g(x) = x2 x 2 x 2 8. g(x) = x3 2x 2 x + 2 x 2 9. f ( x) = 4x2 2x + 1 x 1 y = ( x 1) 2 x 1 ( )( x + 2) y = x 1 ( x 1) 2 Tuesday: 1. Warm up-6 trig functions 2. Due Thursday:! -limits # A circle is inscribed in an equilateral triangle and a square is inscribed in the circle. If the radius of the circle is 10, find the ratio of the area of the square to the area of the equilateral triangle.

50 Graph the 6 trig functions lim f ( x) End behavior limits: x ( ) 2 ( x 2 ) 3x 2 lim x x 3 x + 5 lim x lim ( 2 x ) x =0 ( ) 2 ( 3x 2) 2 x 2 =0 lim ( ) x 7x 3 ( 5 x) ( sin x) lim x 3 ( ) = 3 = 9x x 4 = Continuity (nice enough) of common functions:

51 3x 2, x < 4 21) f(x) is continuous everywhere. f (x) = d, x=4 x 2 + c, x > 4 Find the following. a) the value of c b) lim f (x) x 10 c) the value of d Find lim x 1 + ( Cos 1 (x))

52 Evaluating lim x a ( ) ( ) f x g x Dealing with 0. Find a common factor to cancel! 0 lim x 0 ( 5 + x) 2 25 x 4 16 x lim x 0 x lim x x x

53 lim x 5 5 x 25 x 2 x 2 25 lim x 5 + 2x 2 13x + 15 lim x x 5 3 x 27. a) lim x 5 x 5 x 5 b) lim x 5 + x 5 x 5 c) lim x 5 x 5 x 5 x 5 32 lim x 2 x 2 x 3 + x 2 + 3x + 3 lim x 2 x 4 + x 3 + 2x + 2 Dealing with notzero 0

54 a) lim ( x 3) 2 b) lim x 3 1 ( x 3) 2 c) lim x 3 x ( x 3) a) lim x 5 x 5 x + 5 b) lim x 5 + x 5 x + 5 x 5 c) lim x 5 x + 5 Tuesday: 1. Groups-10 minutes to review limits # Test on Rational functions and limits-thursday 3. A few more thoughts on limits and continuity 4.. HW due Wednesday: 2.1 #1-43 EOO 2.2 #1-49 EOO 2.3 #1-29 EOO 2.4 #1-21 EOO Formal definition of a limit! lim x a f x ( ) = L if

55 lim sin x Evaluate x 0 x = 1 using your calculator. *This is one of the most important limits in calculus and can be proven with the squeeze/ sandwich theorem Use properties of limits to evaluate: lim x 2x + sin x x

56 Use knowledge of end behavior to evaluate: lim x sin x 2x 2 + x Types of discontinuity:

57

58 Properties of continuous functions

59 Find all points of discontinuity of the function. Calculate the limit. lim x 7 3x + 2 2x 3x + 2 lim x 0 + 2x lim x 0 3x + 2 2x

60 lim x 0 lim x 2 3 3x + 2 2x 3x + 2 2x lim x x + 2 2x Composites of continuous functions are continuous: Show that y = xsin x x is continuous Find all points of discontinuity for y = ln( x +1) Tangent lines and rates of change

61 Find the average rate of change of f ( x) = x 3 x over the interval [1,3]. Goal: to perform basic operations on vectors -add, subtract, mult by scalar Key terms for vectors: Vector quantity Scalar quantity Vector addition Resultant Vectors Vector vs scalar quantities: Vectors quantities-magn. and direction velocity acceleration displacement force Scalar quantities- just a magnitude area volume distance direction magnitude speed rate of change

62 Definition of vector quantity Two vectors are the same if they have the same magnitude and direction *length and direction Vector Addition-head to tail Parallelogram method of addition:

63 *you can slide vectors Why? *Notice that addition is commutative (left diagram) Opposite vectors: v vs v Vector Subtraction:

64 Why? Multiplication by scalar:

65 12-2 Algebraic (rectangular) representation of vectors Goal: To use coordinates (components) to perform vector operations instead of the head to tail method (law of cosines) Polar form of a vector example: 12N at 30º Component form of that same vector?

66 Example: put 21 mph, 130º into component form Example put v = ( 4N, 7N ) into polar form. Compass direction.

67 Adding Vectors in component form: ( a,b) + ( c,d) = ( a + c,b + d) Scalar multiplication in component form: k a,b ( ) = ( ka,kb) *makes sense, right?

68 a r refers to value of the magnitude (length of the vector)

69 a uuuu + b r r r a + b uuuu r r r a + b When would = a + b? Example-force Two forces are acting on an object: F 1 is 20 N east and F 2 is 10 N at 150º (compass direction) Find the magnitude and direction of the resultant. Example-displacement pg. 424 #15 Boat 3 mi course of 040º then 8 mi course 100º Distance and course from start to finish? Example- Given A(4,2) and B(9, 1), uuu r a. express AB in component form. uuu r b. express BA in component form uuu r c. Find AB uuu r d. Express AB in polar form

70 Example-finding the resultant Two forces. F 1 is 13 N at 100º. F 2 is 7 N at 170º. Find the resultant... a. Using polar form (LOC) b. Using components Example-physics-ramp problems A 50 lb box sits on a ramp hat makes a 20º angle with the horizontal. a. Find the component of the 50 lb force that is parallel to the ramp b. Find the component of the 50 lb force that is perpendicular to the ramp c. How many lbs of force are needed to prevent the object from sliding down the ramp? -ignore friction, etc. Unit vector-vector with magnitude of... Example- write v=(3, 5) using unit vectors 12-3 Vector and Parametric Equations: Motion in a Plane Goal: To use vector and parametric equations to describe motion in the plane Vector functions-takes in one or more variables and returns a vector

71 *we ll focus on functions that take in one variable Example: Graph v(t) =< t,1 > *How does this differ from the related cartesian function. y=1 -direction -speed - Example: Graph v(t) =< cost,sint > Example: Graph v(t) =< 6 cost, 3sint > Vector equation for an object moving at a constant velocity (a line!): ( x, y) = ( x 0, y 0 ) + t ( a,b) (x,y)=(2,5)+t(-1,6) (x,y)=(2,5)+t( 2,12) m=6/-1 k(-1,6)

72 x 0, y 0 ( ) represents the starting position ( a,b) represents the velocity vector in component form speed = a 2 + b 2 Example: Find a THE vector equation of the line through points ( 3,4 ) and ( 5,5). Label points for specific values of t. Example: Find a different vector equation of the line through points ( 3,4 ) and ( 5,5). Object is moving with a constant velocity. 5,3 ( ) at t=0 (Start)

73 ( 4,15) at t=3 a. Find the velocity (component form) b. Find the velocity (polar) c. Find the speed d. Write a vector equation to describe the motion e. Write the cartesian equation of the path of the object f. When (if ever) does the object reach ( 16, 31) g. When (if ever) does the object reach (10, 13) h. Write the parametric equations for the situation g. Write the cartesian equation of the line by eliminating the parameter Every line has vector/parametric equation(s). Every linear vector/parametric equation(s) describes line(s). Example: An object has parametric equations x = 1 t y = 1+ 2t a. Write the cartesian equation of the line *two methods! b. When and where does the object cross the circle x 2 + y 1 *two methods! ( ) 2 = 25? Example At time t, the position of an object moving with constant velocity is given by the parametric equations

74 x = 2 3t y = 1+ 2t a. What are the velocity and speed of the object b. Where and when does it cross the line x + y = 2 c. Where and when does it cross the x-axis d. Write a vector equation for the object Example-two objects in motion! 12-3 #23 A spider and a fly crawl so that their positions at time t (in seconds) are: spider: ( x, y) = ( 2,5) + t ( 1, 2 ) fly: ( x, y) = ( 1,1) + t ( 1,1) a. Do their lines of travel cross? b. Do they collide? *two methods!! Example-12-3 #24 A spider and a fly crawl so that their positions at time t (in seconds) are: spider: ( x, y) = ( 3, 2 ) + t ( 2,1) fly: ( x, y) = ( 1,6 ) + t ( 4, 3) a. Do their lines of travel cross? b. Do they collide? *two methods!!

75 Example-12-3 #26 Given by x = 8cost y = 8sint a. Describe the path of the object b. Where does it start. Which direction does it travel? c. Write the cartesian equation for the object by eliminating the parameter Now, make the object... a. start at 8,0 b. start at 8,0 c. start at 8,0 d. start at 0, 8 ( ) and complete 2 counterclockwise revolutions ( ) and complete 1 clockwise revolutions ( ) and complete 1 counterclockwise revolutions ( ) and complete 2 clockwise revolutions Example- Find a cartesian equation for x = 2cost and y = 5sint x = cos 3t Example-On the calculator, graph y = sin5t

76 Example-12-3 #39 From a point 30 yd directly in front of the goal poasts, a football is kicked at an angle of elevation of 30º with an initial velocity of 60 ft/s. a. Write a set of parametric equations to describe the motion b. The goal post crossbar is 10 ft above the ground. Will the ball pass over the crossbar? *remember, h( t) = gt 2 + v 0 t + h 0 where g = 16 ft / s 2 or g = 9.8 m / s 2 Thursday 1. A couple more parametric examples 2. Review HW Class Ex. #1-8, Written Ex. # #1-17 odd, 21-24, #1-21 odd, 25, 27, 29 (no calc),

77 3. Vector multiplication-dot product 4. HW due Monday: # 1-19, 25 Parallel and Perpendicular Vectors and the Dot Product ( x 1, y 1 ) and x 2, y 2 x 1, y 1 ( ) are parallel if ( ) = ( kx 2,ky 2 ) ( x 1, y 1 ) and ( x 2, y 2 ) are perpendicular if *we call this the dot product (or scalar product) *also called orthogonal Dot product: If v = ( x 1, y 1 ) and u = ( x 2, y 2 ), then v i u = Example-proving perpendicular and parallel If u = (3, 6), v = ( 4,2), and w = ( 12, 6), a. find u i v and v i w. b. show that u and v are perpendicular Properties of the dot product Angle Between two Vectors Proof

78 Example-To the nearest degree, find the measure of the angle between the vectors (1,2) and ( 3,1) Example-In triangle PQR, P=(2,1), Q=(4,7), and R=( 2,4). Find the measure of m P

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80 What if the vectors are parallel? =

81 Applications of the dot product: 1. Are two vectors perpendicular? do the dot product and look for 0 2. Calculate angle between two vectors 3. Quickly determine if angle between vectors is acute or obtuse (without a calc) costheta=positive means acute 4. Find the projection of one vector onto another 5. Calculate work done by a force F acting on an object and displacing it W = F i s where s is displacement of the object

82 Thursday 1. A couple more parametric examples 2. Review HW Class Ex. #1-8, Written Ex. # #1-17 odd, 21-24, #1-21 odd, 25, 27, 29 (no calc), 3. Vector multiplication-dot product 4. HW due Monday: # 1-19, 25 Parallel and Perpendicular Vectors and the Dot Product ( x 1, y 1 ) and ( x 2, y 2 ) are parallel if

83 ( x 1, y 1 ) = ( kx 2,ky 2 ) ( x 1, y 1 ) and ( x 2, y 2 ) are perpendicular if *we call this the dot product (or scalar product) *also called orthogonal Dot product: If v = ( x 1, y 1 ) and u = ( x 2, y 2 ), then v i u = Example-proving perpendicular and parallel If u = (3, 6), v = ( 4,2), and w = ( 12, 6), a. find u i v and v i w. b. show that u and v are perpendicular Properties of the dot product Angle Between two Vectors Proof Example-To the nearest degree, find the measure of the angle between the vectors (1,2) and ( 3,1) Example-In triangle PQR, P=(2,1), Q=(4,7), and R=( 2,4). Find the measure of m P

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85 What if the vectors are parallel?

86 =

87 Vectors in 3D How many regions do the axis break the coordinate plane up into?

88 Pythag in 3D Distance formula in 3D Midpoint formula in 3D

89 Example-vector given two points If A(1, 4,7) and B( 4,10,8), uuu r a. find AB. uuu r b. find AB Example-equation of a sphere Find the center and radius of the sphere with equation x 2 + y 2 + z 2 6x +10y + 2z = 65 Example-equation of a line in 3D Find the vector and parametric equations of the line containing (2,3,1) and (5,4,6) *there are many answers!! Example-Find a vector equation of the line through (3,9, 5) and parallel to the line L with equation x, y, z ( ) = ( 1,5, 2) + t ( 4,3,2).

90 Two vectors are parallel provided that their velocity vectors are proportional Example-Find the measure of the angle between ( 4, 5,3) and ( 7,0, 1). Example-what is it? Let A and B be fixed points and P is an arbitrary point. Describe the set of points: in 2d? in 3d? a. PA uuu r = PB uuu r b. PA uuu r i PB uuu r = 0 c. PA uuu r = 10 d. PA uuu r + PB uuu r = 10 Wednesday 1. HW quiz 2. Determinants 3. Cramer s rule 4. HW due Thursday #1-3, #3,4, 17, 18 -read 12-9

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95 12-9 cross products Remember... Write ( 3, 5, 9) in unit vector form Planes in 3D: Sketch the following planes z = 2 y = 3 y = x (notice this isn t a line!) 2x + 3y + 4z = 12 (use intercepts) (0,0,3)

96 (0,4,0) (6,0,0) Example-cross products, etc. Let u = ( 4,0,1), v = ( 5 1,0 ), and w = ( 3,1, 2 ) a. v xu = b. Use dot product to confirm that the vector from part a is perpendicular to u and v c. u x v =

97 d. u x v = *use u v sinθ (you will need dot product to find the angle) w = ( 3,1, 2 ) e. Find a vector perpendicular to the plane of u and w. f. Find a vector (there are an infinite amount) perpendicular to just w. g. Find the area of the parallelogram determined by u and v. h. Find the area of the triangle determined by u and v. cross product = (,, ) mag. 1/2 Example-planes defined by 3 points Find a vector perpendicular to the plane determined by P(1,1,0), Q( 1,0,2), and R(2,1,1).

98 Example-name two ways to find the angle between (1,2,2) and (4,3,0) Example-two ways to show if (5, 1,2) and (7, 2,3) are parallel? Example-area of a triangle 3 ways A(11,17,0), B( 8,12,0), and C(25,13,0). Find the area of triangle ABC using... a. cross products b. using A = 1 absinc and dot products 2 c. using Hero s formula

99 Applications of the cross product: `

100 Applications of the dot product: 1. Are two vectors perpendicular? do the dot product and look for 0 2. Calculate angle between two vectors i. Quickly determine if angle between vectors is acute or obtuse (without a calc) costheta=positive means acute 4. Find the projection of one vector onto another 5. Calculate work done by a force F acting on an object and displacing it W = F i s where s is displacement of the object

101 Tuesday: 1. Check/review hw 2. More transformations -rotations 3. Three ways to solve systems -row operations -cramer s rule -inverses 4. HW due Tuesday #5-10, 34, 35 (use inverses to solve) #33, 35 (use inverses) Matrices-basics 1. Operations -add -subtract -multiply (not commutative) -dot product -we don t divide (inverses to solve AB=C) -transpose (switch rows and columns -calculate inverse -if 2x2 matrix, use determinant -otherwise, row operation method (not in this class) Applications of Matrices:

102 1. Transformations -translations use addition -otherwise, multiplication 2. Organizing data where dot products make sense 3. Solving systems of equations with matrices -row operations (not in this class) -cramer s rule -inverses matrices (using determinant) Transformation Matrices Theorem-the columns of a transformation matrix are the images of the points (1,0) and (0,1) respectively. Proof: T = a b c d a b 1 c d 0 = a b 0 c d 1 =

103 Summary of matrix transformations: Write each transformation of (x,y) in T: form and in matrix form In words ( ) a b T: x, y reflection over x-axis (x, y) reflection over y-axis reflection over y=x left 3 up 5 rotation clockwise 90º about the origin rotation 180º about the origin rotation counterclockwise 90º about the origin rotation θ about the origin c d

104 Example-transforming several points Consider T :( x, y) ( 2x, y) a. Write a transformation matrix b. Set up a product of matrices to represent the transformation of the points (1,5), ( 2, 0), and (6,4). c. describe the transformation in words d. describe using function notation e. write as two separate transformations Example-translations Describe each translation a. G :( 3, 1) ( 8, 2) b. x + 6 = x' y 1 = y' c. x y = x' y' Express the translation as a matrix equation:

105 P( 2, 4);P' ( 3,1) Compositions of Transformation Matrices Example- Calculate R 90º R 90º ( ) 2 Calcuate R 90º Example- Examplecalculate XY where... X =reflection over x-axis Y= reflection over y-axis Examplecalculate LX where L is a reflection in the line y=x and X is a reflection over the y-axis

106 Examplecalculate LM where L is a reflection over y=x and M is a reflection over y= x Example-Find a 2x2 matrix that projects a point onto the x-axis (keeps x-coordinate and makes y-coordinate 0). Why does this matrix not have an inverse? Example-find a matrix that rotates a point (x,y) through an angle of θ Example- test the above for rotations of 90º and 180º Example- The 3 Ways to Solve Systems with Matrices

107 Example-using row operations to solve systems (not on test or final exam) Use row operations to solve the system:

108 Example-solve using cramer s rule Solving using inverses

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110 Example-finding and checking an inverse

111 Example-order matters Solving AX = B vs XA = B Example-solve using inverses Example-solving matrix equation (including addition)

112 *there are two ways to set up this matrix equation Example- no solutions

113 2x + 5y = 7 4x +10y = 1 Example-all reals 2x + 4y = 7 2x + 8y = 14 Example-determine which matrix product AB or BA makes sense. What does the product tell you? Let A=

114 Let B=